Pride keeps most people from saying those four little words: "It's all my fault." The most difficult part of repairing damaged relationships is getting the person on either side of the conflict to admit guilt. With careful phrasing, however, you can make up with someone you care about without admitting you did anything wrong. Simply apologise for the effects without naming yourself as the cause. For example, say "I'm so sorry you fell..." instead of "I'm so sorry I pushed you."
Organise your thoughts. Decide why you wish to write a letter of apology if you have chosen not to admit guilt. Are you sorry that a conversation turned into an argument? Are you sorry that you and a loved one are not speaking because of a disagreement? Are you sorry because an event has caused your work environment to feel tense and uncomfortable? Know exactly why you are apologising before you begin to write.
Address the recipient using the word "Dear," even if the letter is for a professional colleague. If the letter is professional in nature, use the recipient's first and last names. If the letter is personal, only use the person's first name.
Begin with appreciation. Tell the recipient why you love her/like him/enjoy working with her. Point out his strengths and how he has enhanced your life. If you have quarrelled with your wife, for example, begin with "I love you. I'm so happy you married me." If you've argued with a co-worker, write, "I appreciate how your expertise and insights have assisted my career."
Be truthful with your apology. For example, if you are sorry because your wife is not speaking to you, say so. If you are sorry because your co-worker has not eaten lunch with you since the argument, say so. If you are sorry that something you did caused a loved one pain, admit it. Consider statements like these: "I'm sorry we argued. I miss your voice. I miss spending time with you. I hate the way our disagreement has affected our relationship. I'm sorry you are hurting."
Propose a compromise that allows you to take responsibility for your behaviour. For example, "I'd like to talk about this again and try to arrive at a solution that benefits us both. I promise to listen with an open mind and without interrupting you. I promise to speak without raising my voice. I promise to speak to you in a way that reflects how much I love/like/respect you, even if I disagree with your point."